Complex random matrices and Rician channel capacity

Abstract

The eigenvalue densities of complex noncentral Wishart matrices are investigated to study an open problem in information theory. Specifically, the largest, smallest and joint eigenvalue densities of complex noncentral Wishart matrices are derived. These densities are expressed in terms of complex zonal polynomials and invariant polynomials. The connection between the complex Wishart matrix theory and information theory is given. This facilitates the evaluation of the most important information-theoretic measure, the so-called ergodic channel capacity. In particular, the capacity of multipleinput, multiple-output (MIMO) Rician distributed channels is investigated. We consider both spatially correlated and uncorrelated MIMO Rician channels and derive the exact and easily computable tight upper bound formulas for ergodic capacities. Numerical results are also given, which show how the channel correlation degrades the capacity of the communication system. Key words. complex random matrix, complex noncentral Wishart matrix, zonal polynomial, invariant polynomial, Rician distributed MIMO channel, ergodic channel capacity AMS(MOS) subject classification. 94A15, 94A05, 60E05, 62H10To appear in Problems of Information Transmission. RésuméOnétudie la densité des valeurs propres de matrices de Wishart noncentrales complexes en vue d'applicationsà un problème ouvert de la théorie de l'information. On obtient la densité conjointe de la plus grande et de la plus petite de ces valeurs propres au moyen de polynômes zonaux complexes et de polynômes invariants. Onétablie la relation entre ces matrices et la théorie de l'information pourévaluer la très importante capacité ergodique d'un canal. Onétudie la capacité des canaux riciens distribuésà entrées et sorties multiples (MIMO). On considère les canaux correlés et non correlés dans l'espace et l'on obtient facilement la borne supérieure exacte des capacités ergodiques. Les résul-tats numériques montrent que la corrélation du canal réduit la capacité du système de communication.

Matricvariate and matrix multivariate T distributions and associated distributions

“…Now, we show three Jacobians in terms of the β parameter, which are based on the work of Kabe [20] and Dimitriu [11]. These results are proposed as extensions of real, complex or quaternion cases, see James [19], Khatri [22], Metha [26], Ratnarajah et al [30] and Li and Xue [25], also see Díaz-García and Gutiérrez-Jáimez [6].…”

“…The complex case has renewed interest in multivariate analysis in diverse areas of science and technology, see Mehta [29], Ratnarajah et al [33] and Micheas et al [30], among many others. Moreover, diverse works involving multivariate analysis have appeared in the context of the quaternion case, see Bhavsar [2], Forrester [17], Li and Xue [28], among others.…”

Analysis of Optimal Combining in Rician Fading With Co-Channel Interference

“…This case, where Ψ is assumed to be an identity matrix, is widely discussed in the literature. The eigenvalue distribution of this case, i.e., a non-central Wishart matrix with a covariance matrix Σ which is not an identity matrix, is analyzed in [32] in terms of zonal polynomials. However, using this eigenvalue distribution to obtain the Laplace transform expression of η becomes mathematically intractable.…”

Generalisations of some properties of invariant polynomials with matrix arguments

“…However, important distributional problems cannot be solved via zonal polynomials, such as the distribution of the eigenvalues of a noncentral Wishart distribution or the doubly noncentral Beta type I and II distributions. Solutions of such problems have been provided via invariant polynomials with matrix arguments; see for example Davis (1980), Chikuse (1980), Chikuse (1981), Chikuse and Davis (1986), Díaz-García and Gutiérrez-Jáimez (2001) and Ratnarajah et al (2005), among many others.…”

“…; see Davis (1980), Chikuse and Davis (1986) for the real case, and Ratnarajah et al (2005) for the complex case.…”